Supporting Information Corrected June 17, 2015 Supporting Information: Materials and Methods, Results and Discussion Figures S1-S4 Tables S1-S10 References (1-26) SUPPORTING INFORMATION 1. SUPPORTING MATERIALS AND METHODS 1.1 Assumptions about the Isthmus of Panama Hundreds of researchers have assumed a closure date for the Isthmus of Panama between 3.5 and 3.0 Ma over the last two decades. The papers most cited for this geological date are essentially from two research groups, Coates (1-5) and Haug (6, 7). Taken together, these papers alone have been cited 2092 times, according to Google Scholar (accessed March 4, 2015). 1.2 Molecular data compilation and analyses We used molecular phylogenetic studies that included extant New World species published or in press until May 2013 in 29 journals (Tables S8 and S9). We identified 424 divergence time estimates in 169 phylogenies that represented unambiguous instances of dispersal across the Isthmus of Panama or vicariance events of marine taxa across the Caribbean and Pacific oceans. Our proxy for dispersal time, mean crown ages S-1 and the confidence intervals around them, were gleaned from each phylogeny. Crown groups are defined as the least inclusive monophyletic group that contains all the extant members of a clade (sensu 8). We adopted the approach of Hoorn et al. (9) for data mining and included studies that were directly or indirectly dated with fossil calibration or known rates of nucleotide substitution and excluded studies with calibrations based on geologic events, such as the closure of the Panama Isthmus. 1.3 Fossil data generation To compile fossil mammal data, searches were conducted in Web of Knowledge (http://wokinfo.com) and Georef (http://www.agiweb.org/georef/). The key terms used in the searches included: fossil mammal(s), South America, geological epochs from the Paleocene to the Pleistocene, and the names of all South American countries. New fossil occurrences were added to the Paleobiology Database (PBDB; http://paleodb.org/). From this database, a total of 32,342 fossil mammal records were gathered for this study. We assigned the origin of each fossil mammal occurrence to its respective family or order using information from the fossil record and the literature. 1.4 Fossil data analysis All undetermined fossil occurrences and all records that had not been identified to species level were removed, leaving 23,090 occurrences in the analyses. The final data set (available from http://datadryad.org) included 3,589 species, of which 2,657 (21,104 occurrences) were found in North America and 958 (1,986 occurrences) in South America (Fig. 2A). We classified each fossil occurrence as North or South American if S-2 the taxon or its ancestor(s) were in either North or South America before 10 Ma following Carillo et al. (10) and identified 50 South American clades in North America (494 occurrences) and 182 North American lineages in South America (439 occurrences). All fossil occurrences were provided with minimum and maximum ages, derived from the temporal ranges of the respective geological units and reflecting the degree of uncertainty in their dating. These were treated as confidence intervals and used as uniform ranges from which the ages of the occurrences were randomly drawn (11). The ages of the fossil occurrences of each species were randomly resampled 1,000 times from the respective ranges and used to determine the first and last appearances of species, while accounting for the dating uncertainties of the fossil record. Based on the first and last appearances, we calculated the raw number of species in North and South America within time bins of 1 Myr and plotted diversity trajectories through time with 95% confidence intervals obtained from the 1,000 randomized samples (Fig. 3B, Table S2). Based on these counts the proportions of migrant species in the two continents were also quantified for each time bin (Fig. 2C, Table S2). 1.5 Re-estimation of divergence times A number of marine lineages exhibit trans-Isthmian divergences that are assumed to have taken place concomitantly with the final closure of the Isthmus of Panama (reviewed in 12). The genetic distances among these geminate species pairs have been used to calibrate molecular clocks based on a closure of the Isthmus of Panama between 3.5 and 3.0 Ma. This molecular clock has then been used to estimate the divergence times of S-3 sister pairs that occur in the marine ecosystems on each side of the Isthmus (e.g. 13, 14, 15). Because we did not include molecular dates calibrated with any estimate of Isthmus closure, we had to estimate divergence dates for marine taxa using an independent rate in order to include these taxa in our study. We obtained genetic distances (Kimura’s two-parameter distance) for trans-isthmian species pairs from Lessios (25) and applied a standard molecular clock rate of 2% mitochondrial sequence divergence per million years to convert values to absolute time. This molecular clock has substantial uncertainty surrounding it, thus divergence times for these species pairs should thus be considered as coarse approximations. Although molecular dating based on secondary rate calibrations is prone to potential caveats (e.g. 16, 17), re-estimation allowed for a comprehensive synthesis on trans-isthmian divergences in both marine and terrestrial systems. 1.6 Estimating migration rates from dated migration events Migration events (dispersals and vicariances) extracted from dated phylogenies were used to estimate rates of migrations and their variation through time. Migrations were treated as independent events occurring through time according to a random Poisson process. In our notation, the ages of all events are measured as time before the present. For a number K of migration events, we define a vector of times of migration t = t1, t2,... tK, tK+1, where t1 and tK represent the oldest and most recent events respectively, and tK+1 = 0 represents time 0, i.e. the present. Assuming that the rate parameter changes over time according to a function λ(t), the likelihood of t based on a non-homogeneous Poisson process is: S-4 K tj+1 K P (t ,...t λ)= exp λ(u) du λ(t ) 1 K | − j j=1 tj ! j=1 Y Z Y , (1) where the first product describes the probability of the waiting times between events and the second product accounts for each migration events based on the variable rate λ. Note that tK+1 appears in the first product where it accounts for the waiting time between the last migration event and the present (i.e. tK – tK+1), whereas it is not included amongst the migration events in the second product. Several Poisson models were implemented, the simplest of which had one constant rate through time, i.e. λ(t) = λ, for t1 ≤ t ≤ 0, that is a single free parameter estimated from the data (degrees of freedom, d.f. = 1). A model with exponentially distributed rate was generated using the following function: λ(t)=q + l exp( lt) − , (2) where q ≥ 0 is a minimum baseline rate and l is the rate parameter of the exponential distribution. Both q and l were estimated from the data as two independent free parameters (d.f. = 2). Because the times of migration are expressed as time before present (t = 0 being the present), this function models a rate that exponentially increases towards the present. To allow for more complex patterns of rate variation, models with rate shifts through time were additionally implemented. These assumed that rate changes occurred S-5 at times of shift τ defining time frames, while the rates were assumed to be constant within each time frame. For a set of T times of shift defining T+1 time frames, a vector of rates λ 1, … λ T+1 was defined and used to calculate the probability of Equation (1). Each rate shift introduces two additional parameters in the model, i.e. the time of shift and the migration rate following the shift. For instance, a model with one rate shift has two rate parameters { λ 1, λ 2 } and one time of shift { τ1 } (d.f. = 3). Implementation - The fit of alternative models of migration were compared in a maximum likelihood framework. Because the sample size in several of the data sets was comparatively small (Table S10), we used the second-order Akaike Information Criterion (AICc; 18, 19) in all model comparisons, and AICc thresholds determined through simulations (see below) as indicators of significant support for a more parameter-rich model against a simpler model. We used a stepwise procedure that starts with a constant rate model and subsequently adds rate shifts to determine the best number of rate shifts supported by the data (e.g., 20, 21). At every step, the algorithm adds a rate shift in the model, estimates the parameters (migration rates and times of shift) by likelihood optimization, and computes the AICc score. If the AICc for this model is significantly lower (i.e. better) than the previous score, based on the AICc threshold, the model is retained and one additional rate shift is tested. The procedure is stopped when an additional rate shift does not yield a significant improvement of AICc score, in which case the previous (simpler) model is retained (20). For model comparison, we also ran maximum likelihood optimizations under two models with fixed times of rate shift, which we named the ‘Standard’ and the ‘Miocene’ models following Coates and Stallard S-6 (5). The Standard model assumed a shift at 3.5 Ma and two rate parameters (before and after the shift; d.f. = 2) and the Miocene Model had shifts at 25 and 15 Ma and three rate parameters (d.f.